Supersymmetric heterotic solutions via non-SU(3) standard embedding

A supersymmetric solution to type II supergravity is constructed by superposing two hyperk\"ahlers with torsion metrics. The solution is given by a K\"aler with torsion metric with $SU(3)$ holonomy. The metric is embedded into a heterotic solution obeying the Strominger system, together with a Yang-Mills instanton obtained by the standard embedding. T dualities lead to an $SO(6)$ instanton describing a symmetry breaking from $E_8$ to $SO(10)$. The compactification by taking a periodic array yields a supersymmetric domain wall solution of heterotic supergravity.